The image shows a problem where two lines $y = \frac{1}{2}x - 2$ and $y = -2x + 8$ intersect at point $(4, 0)$, along with the vertical line $x = -2$. We are asked to find the area of the triangle formed by these two lines and the line $x = -2$.

Steps to solve:

1. Find the intersection points:

   • We already know that the lines intersect at $(4, 0)$ (this is where $y = 0$).
   • The second line, $x = -2$, intersects both lines as well.

2. Find intersection of $y = \frac{1}{2}x - 2$ and $x = -2$:

   • Substitute $x = -2$ into $y = \frac{1}{2}x - 2$: $y = \frac{1}{2}(-2) - 2 = -1 - 2 = -3$
   • So the point of intersection is $(-2, -3)$.

3. Find intersection of $y = -2x + 8$ and $x = -2$:

   • Substitute $x = -2$ into $y = -2x + 8$: $y = -2(-2) + 8 = 4 + 8 = 12$
   • So the point of intersection is $(-2, 12)$.

4. Coordinates of the triangle: The triangle is formed by the points:

   • $(-2, 12)$
   • $(-2, -3)$
   • $(4, 0)$

5. Calculate the area of the triangle: Use the formula for the area of a triangle given three vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$: $\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$ Substituting the points $(-2, 12), (-2, -3), (4, 0)$: $\text{Area} = \frac{1}{2} \left| -2(-3 - 0) + (-2)(0 - 12) + 4(12 + 3) \right|$ $\text{Area} = \frac{1}{2} \left| -2(-3) + (-2)(-12) + 4(15) \right|$ $\text{Area} = \frac{1}{2} \left| 6 + 24 + 60 \right| = \frac{1}{2} \times 90 = 45$

Thus, the area of the triangle is 45 square units.

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. How do you find the area of a triangle using the determinant method?
2. How do you determine the intersection point of two lines?
3. What is the significance of the slope in line equations?
4. How would the area change if one of the lines had a different slope?
5. How can you find the area of a polygon with more than three sides using vertices?

Tip: When solving geometry problems involving triangles, always plot the points to visualize the shape and verify your calculations.